Exact confidence regions for scale and location parameters based on sample quartiles

In this paper we present relatively simple (ruler, paper, and pencil) nonparametric procedures for constructing joint confidence regions for (i) the median and the inner quartile range for the symmetric one-sample problem and (ii) the shift and ratio of scale parameters for the two-sample case. Both procedures are functions of the sample quartiles and have exact confidence levels when the populations are continuous. The one-sample case requires symmetry of first and third quartiles about the median.

The confidence regions we propose are always convex, nested for decreasing confidence levels and are compact for reasonably large sample sizes. Both exact small sample and approximate large sample distributions are given.     

Bayesian inference for joint location and scale nonlinear models with skew-normal errors

A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the dataset under consideration involves asymmetric outcomes. In this article, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis for joint location and scale nonlinear models with skew-normal errors, which relax the normality assumption and include the normal one as a special case. The main advantage of these class of distributions is that they have a nice hierarchical representation that allows the implementation of MCMC methods to simulate samples from the joint posterior distribution. Finally, simulation studies and a real example are used to illustrate the proposed methodology.     

Asymptotic power efficiency for a location and scale problem

The asymptotic power efficiency of the class of linear rank tests relative to the asymptotically most powerful rank test is derived for a two sample location and scale problem and numerical evaluations are presented for two special tests.     

Randomized confidence intervals of a parameter for a family of discrete exponential type distributions

For a family of one-parameter discrete exponential type distributions, the higher order approximation of randomized confidence intervals derived from the optimum test is discussed. Indeed, it is shown that they can be asymptotically constructed by means of the Edgeworth expansion. The usefulness is seen from the numerical results in the case of Poisson and binomial distributions.     

Confidence intervals for the ratio of two variances

In this paper we consider confidence intervals for the ratio of two population variances. We propose a confidence interval for the ratio of two variances based on the t-statistic by deriving its Edgeworth expansion and considering Hall's and Johnson's transformations. Then, we consider the coverage accuracy of suggested intervals and intervals based on the F-statistic for some distributions.     

A Wavelet Method for Confidence Intervals in the Density Model

Abstract.  We present a wavelet procedure for defining confidence intervals for f ( x ₀), where x ₀ is a given point and f is an unknown density from which there are independent observations. We use an undersmoothing method which is shown to be near optimal (up to a logarithmic term) in a first order sense. We propose a second order correction using the Edgeworth expansion. The adaptation with respect to the unknown regularity of f is given via a Lepskii type algorithm and has the advantage to be well located. The theoretical results are proved under weak assumptions and concern very irregular or oscillating functions. An empirical study gives some hints for choosing the constant of the threshold level. The results are very encouraging for the length of the intervals as well as for the coverage accuracy.     

Higher order moments of order statistics from the Lindley distribution and associated inference

ABSTRACT

The Lindley distribution is an important distribution for analysing the stress–strength reliability models and lifetime data. In many ways, the Lindley distribution is a better model than that based on the exponential distribution. Order statistics arise naturally in many of such applications. In this paper, we derive the exact explicit expressions for the single, double (product), triple and quadruple moments of order statistics from the Lindley distribution. Then, we use these moments to obtain the best linear unbiased estimates (BLUEs) of the location and scale parameters based on Type-II right-censored samples. Next, we use these results to determine the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics to develop Edgeworth approximate confidence intervals of the location and scale Lindley parameters. In addition, we carry out some numerical illustrations through Monte Carlo simulations to show the usefulness of the findings. Finally, we apply the findings of the paper to some real data set.     

Linear estimation of the location and scale parameters based on selected order statistics

This paper gives a review of the best linear estimates of the location and/or scale parameters based on a few order statistics selected from a complete or censored sample. Small sample and large sample cases are considered and compared. Some examples of the practical applications of the estimates are outlined.     

Parametric inference for quantile event times with adjustment for covariates on competing risks data

ABSTRACT

We propose parametric inferences for quantile event times with adjustment for covariates on competing risks data. We develop parametric quantile inferences using parametric regression modeling of the cumulative incidence function from the cause-specific hazard and direct approaches. Maximum likelihood inferences are developed for estimation of the cumulative incidence function and quantiles. We develop the construction of parametric confidence intervals for quantiles. Simulation studies show that the proposed methods perform well. We illustrate the methods using early stage breast cancer data.     

Objective Bayesian inference for the ratio of the scale parameters of two Weibull distributions

The Weibull distribution is widely used due to its versatility and relative simplicity. In our paper, the non informative priors for the ratio of the scale parameters of two Weibull models are provided. The asymptotic matching of coverage probabilities of Bayesian credible intervals is considered, with the corresponding frequentist coverage probabilities. We developed the various priors for the ratio of two scale parameters using the following matching criteria: quantile matching, matching of distribution function, highest posterior density matching, and inversion of test statistics. One particular prior, which meets all the matching criteria, is found. Next, we derive the reference priors for groups of ordering. We see that all the reference priors satisfy a first-order matching criterion and that the one-at-a-time reference prior is a second-order matching prior. A simulation study is performed and an example given.     

Location and scale parameter estimation from randomly censored data

The problem of location and scale parameter estimation from randomly censored data is analyzed through use of a regression model for the Kaplan-Meier quantlle process. Continuous time regression techniques are employed to construct estimators that are both asymptotically normal and efficient. Estimators with a particularly simple form are obtained for the Koziol-Green model for random censorship. In the event of no censoring the regression model, and resulting estimators, reduce to those proposed by Parzen (1979 a, b).     

Matching pseudocounts for interval estimation of binomial and Poisson parameters

ABSTRACT

For interval estimation of a binomial proportion and a Poisson mean, matching pseudocounts are derived, which give the one-sided Wald confidence intervals with second-order accuracy. The confidence intervals remove the bias of coverage probabilities given by the score confidence intervals. Partial poor behavior of the confidence intervals by the matching pseudocounts is corrected by hybrid methods using the score confidence interval depending on sample values.     

Comparison of tests for bivariate symmetry versus location and/or scale alternatives

This paper studies the performance of tests which use a null hypothesis of bivariate symmetry and detect the broad class of location and/or scale alternatives . The conditionally distribution-free tests of Sen (1967) and Hollander (1971) and parametric tests related to those of Bell and Haller (1969) are compared in a Monte Carlo study which also includes a new conditionally distribution-free test.     

On the robustness of empirical likelihood ratio confidence intervals for location

The authors examine the robustness of empirical likelihood ratio (ELR) confidence intervals for the mean and M‐estimate of location. They show that the ELR interval for the mean has an asymptotic breakdown point of zero. They also give a formula for computing the breakdown point of the ELR interval for M‐estimate. Through a numerical study, they further examine the relative advantages of the ELR interval to the commonly used confidence intervals based on the asymptotic distribution of the M‐estimate.     

Exact likelihood inference for two exponential populations based on a joint generalized Type-I hybrid censored sample

Following the work of Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring. Comm Statist Theory Methods. 1988;17:1857–1870], several results have been developed regarding the exact likelihood inference of exponential parameters based on different forms of censored samples. In this paper, the conditional maximum likelihood estimators (MLEs) of two exponential mean parameters are derived under joint generalized Type-I hybrid censoring on the two samples. The moment generating functions (MGFs) and the exact densities of the conditional MLEs are obtained, using which exact confidence intervals are then developed for the model parameters. We also derive the means, variances, and mean squared errors of these estimates. An efficient computational method is developed based on the joint MGF. Finally, an example is presented to illustrate the methods of inference developed here.     

Variable selection in joint location and scale models of the skew-normal distribution

A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the data set under consideration involves asymmetric outcomes. Variable selection is an important issue in all regression analyses, and in this paper, we investigate the simultaneously variable selection in joint location and scale models of the skew-normal distribution. We propose a unified penalized likelihood method which can simultaneously select significant variables in the location and scale models. Furthermore, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the location and scale models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies and a real example are used to illustrate the proposed methodologies.     

Markov chain Monte Carlo exact inference for binomial and multinomial logistic regression models

We develop Metropolis-Hastings algorithms for exact conditional inference, including goodness-of-fit tests, confidence intervals and residual analysis, for binomial and multinomial logistic regression models. We present examples where the exact results, obtained by enumeration, are available for comparison. We also present examples where Monte Carlo methods provide the only feasible approach for exact inference.     

A note on the location parameters of two exponential distributions under order restrictions

The problem of simultaneous estimation of location parameters of two independent exponential distributions is considered when location and/or scale parameters are ordered. We show that the standard estimators in the unrestricted case which use information only from the populations individually can be improved upon when various order restrictions are known to hold. The improved estimators are obtained under the quadratic loss function